23-24高一下·全国·课前预习
1 . 请将复数集、实数集、虚数集、纯虚数集之间的关系用下图表示,并填在合适的空间.
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2 . 复数的定义
我们把形如
的数叫做复数,其中i叫做_________ ,满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bef5a75ca975c523f7233f7a7585f4.png)
_________ .
我们把形如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77df5eaef09da6f7c46b4ab5e1ffe92b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bef5a75ca975c523f7233f7a7585f4.png)
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23-24高一下·全国·课前预习
3 . 对于复数
, 复数
,
为实数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
__________ ;
为虚数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
__________ ;
为纯虚数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
__________ ;
为非纯虚数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
__________ .
即复数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff022b87295c5aa3a81894661f0bf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff022b87295c5aa3a81894661f0bf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff022b87295c5aa3a81894661f0bf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f39c662a3927de39135c3eee4b9cb68f.png)
即复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff022b87295c5aa3a81894661f0bf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4438414f3c7c9694128f5a4a5ab672e.png)
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23-24高一下·全国·课前预习
4 . 在复数集
中任取两个数
,
,规定
与
相等当且仅当____________ ,即复数相等:
⇔______
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6cef8cac4bc6a00c4c0e6fbd99bf30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eed3d568acf369a315c7ab41c081049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e39f8634e9eccb84bce5dd266a241909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eed3d568acf369a315c7ab41c081049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f87e591b738d9cda5084e5743354a514.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4063994562bb3788c4999b82ff82b3d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1c822de0f8922c0cfef29e76d24fb19.png)
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23-24高一下·全国·课前预习
5 . 复数集
全体复数所构成的集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8030bb28d45a7fbf9fbf03730c654146.png)
__________ 叫做复数集.
全体复数所构成的集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8030bb28d45a7fbf9fbf03730c654146.png)
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23-24高一下·全国·课前预习
6 . 复数通常用字母z表示,即
,其中的a与b分别叫做复数z的______ 与________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbff022b87295c5aa3a81894661f0bf0.png)
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23-24高一下·全国·课前预习
7 . 复数的乘法
(1)复数的乘法法则
设
是任意两个复数,那么它们的积![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d355a0ab521aef48518679cf44ea27e7.png)
______________ .
(2)复数乘法的运算律
对于任意
,有
(1)复数的乘法法则
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7061314deb8ce88af5e218709a4ea17c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d355a0ab521aef48518679cf44ea27e7.png)
(2)复数乘法的运算律
对于任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7dc24ba4bc1cae38927fcbce43e2419.png)
交换律 | ![]() |
结合律 | ![]() |
乘法对加法的分配律 | ![]() |
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23-24高一下·全国·课前预习
8 . 复数加、减法的几何意义
如图,设在复平面内复数
对应的向量分别为
,以
为邻边作平行四边形,则与
对应的向量是_________ ,与
对应的向量是_____________ .
如图,设在复平面内复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d860cb86e1467ac24010aecfc7a425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a9e798dc024cf6d6815eb96a1f97b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95aa323fbc002e36c732051858419868.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34b88f343ca5a4c29057465541b9cf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e4e2b866b0043a32fc78326553841d3.png)
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23-24高一下·全国·课前预习
9 . 共轭复数
(1)定义:当两个复数的实部_________ ,虚部____________ 时,这两个复数叫做互为共轭复数.虚部不等于0的两个共轭复数也叫做共轭虚数.
(2)表示方法:复数z的共轭复数用
表示,即如果
,那么![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545141bbe7cc66d18946a129179d8df1.png)
__________ .
(1)定义:当两个复数的实部
(2)表示方法:复数z的共轭复数用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fcfebd9f5a57036e6df6b6e14865da3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545141bbe7cc66d18946a129179d8df1.png)
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